Question

Six vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}, \mathbf{e}, \mathbf{f} \) have the magnitudes and directions indicated in the figure. Which of the following statements is true?

• A. \( \mathbf{b} + \mathbf{e} = \mathbf{f} \)
• B. \( \mathbf{b} + \mathbf{c} = \mathbf{f} \)
• C. \( \mathbf{d} + \mathbf{c} = \mathbf{f} \)
• D. \( \mathbf{d} + \mathbf{e} = \mathbf{f} \)

Hint:

When adding vectors, use the head-to-tail method to find the resultant vector. Ensure the magnitudes and directions align correctly.

Answer 1

The Correct Option is D

To solve this question, we will analyze the vectors based on the given directions and magnitudes. Since the image has specific vector directions and magnitudes, we describe how the vectors relate to each other based on their orientations and use the vector addition principle to find the correct statement.
Step 1: Understanding Vector Addition Vector addition is commutative, meaning the order of addition does not change the result. The sum of two vectors can be calculated geometrically using the head-to-tail method.
Step 2: Analyzing the Given Statements 1. Statement (A): \( \mathbf{b} + \mathbf{e} = \mathbf{f} \) We check if the sum of \( \mathbf{b} \) and \( \mathbf{e} \) results in \( \mathbf{f} \). We need to ensure the direction and magnitude are consistent with \( \mathbf{f} \). 2. Statement (B): \( \mathbf{b} + \mathbf{c} = \mathbf{f} \) We verify if adding \( \mathbf{b} \) and \( \mathbf{c} \) gives \( \mathbf{f} \). This can be done geometrically, checking the vector sum. 3. Statement (C): \( \mathbf{d} + \mathbf{c} = \mathbf{f} \) Similarly, we check if the sum of \( \mathbf{d} \) and \( \mathbf{c} \) results in \( \mathbf{f} \). 4. Statement (D): \( \mathbf{d} + \mathbf{e} = \mathbf{f} \) We also check this sum and see if the vectors \( \mathbf{d} \) and \( \mathbf{e} \) geometrically add up to \( \mathbf{f} \).
Step 3: Conclusion After performing the vector addition and considering the directions and magnitudes, we conclude that the correct option is: \[ \boxed{(D) \, \mathbf{d} + \mathbf{e} = \mathbf{f}} \] This is the only combination where the vectors geometrically align and add up to the correct resulting vector.